The gravitational wave strain emitted by a perturbed black hole (BH) ringing down is typically modeled analytically using first-order BH perturbation theory. In this Letter, we show that second-order effects are necessary for modeling ringdowns from BH merger simulations. Focusing on the strain's (ℓ,m)=(4,4) angular harmonic, we show the presence of a quadratic effect across a range of binary BH mass ratios that agrees with theoretical expectations. We find that the quadratic (4,4) mode's amplitude exhibits quadratic scaling with the fundamental (2,2) mode-its parent mode. The nonlinear mode's amplitude is comparable to or even larger than that of the linear (4,4) mode. Therefore, correctly modeling the ringdown of higher harmonics-improving mode mismatches by up to 2 orders of magnitude-requires the inclusion of nonlinear effects.